Hi again,
I am coming back to the estimation of the covariance. I tried to use your advice in assuming a common shock. In particular, I set my exogenous processes as follows:
v1 = rhoI1v1(1) + rhoI12v2(1) + stdr_ist*(eta1 + PHIeta3) ;
v2 = rhoI1v2(1) + rhoI12v1(1) + stdr_ist(eta2 + PHI*eta3);
Thus, the variance of the “overall” shock is stdr_ist^2 + (stdr_ist^2)(PHI^2) and the covariance is stdr_istPHI (correct??). Then, my shock block writes:
shocks ;
var eta1 = 1;
var eta2 = 1;
var eta3 = 1;
end;
and I do not set anything for the covariance. If I estimate only rhoI1, rhoI12 and stdr_ist (MLE without setting a prior), it converges normally and I get rather logical results:
Improvement on iteration 11 = 0.000000000
improvement < crit termination
Objective function at mode: 169.780830
Objective function at mode: 169.780830
RESULTS FROM MAXIMUM LIKELIHOOD
parameters
Estimate s.d. tstat
stdr_ist 0.0161 0.0014 11.2415
rhoI1 0.9651 0.0022 438.0741
rhoI12 0.0326 0.0031 10.4305
Note 1: Theta is 2
Note 2: Epsilon is 2
Note 3: Interim period is 0
Total computing time : 0h00m09s
However, I am not sure what Dynare assumes as the covariance between the shocks. Does it indeed assume/impose zero covariance?? Trying to estimate PHI becomes a mess, as it is extremely sensitive to the initial condition. If I set the latter to 0.02, say:
estimated_params ;
stdr_ist , 0.0126 ,;
rhoI1 , 0.95 , ,;
rhoI12 , 0.0202 , , ;
PHI , 0.02, , ;
end;
estimation(datafile=istshocks1, mode_compute=4) ; %mode_compute=4
it converges and gives me what seem to be logical results.
Improvement on iteration 303 = 0.000000090
improvement < crit termination
Objective function at mode: 171.298880
Objective function at mode: 171.298880
RESULTS FROM MAXIMUM LIKELIHOOD
parameters
Estimate s.d. tstat
stdr_ist 0.0147 0.0043 3.4273
rhoI1 0.9558 0.0022 441.7531
rhoI12 0.0417 0.0034 12.1793
PHI 0.3894 0.7154 0.5443
Note 1: Theta is 2
Note 2: Epsilon is 2
Note 3: Interim period is 0
Total computing time : 0h00m44s
Nevertheless, even the most minor changes to this initial condition cause trouble. Initial conditions of PHI = 0.019 and 0.0199 (!!) result to the message of no convergence:
POSTERIOR KERNEL OPTIMIZATION PROBLEM!
(minus) the hessian matrix at the “mode” is not positive definite!
=> posterior variance of the estimated parameters are not positive.
You should try to change the initial values of the parameters using
the estimated_params_init block, or use another optimization routine.
Warning: The results below are most likely wrong!
In dynare_estimation_1 at 643
In dynare_estimation at 62
In BS_CapitalServices3IST1 at 1066
In dynare at 132
Setting PHI = 0.019999 does converge but the estimated parametres are a bit different from the case where the initial condition is 0.02.
My questions are:

When I do not set anything for the covariance and I do not ask for the estimation of PHI, does Dynare indeed assume /impose a zero covariance? That is, are the persistence coefficient and the common standard deviation estimated with ML under the restriction of zero covariance?

The issues of nonconvergence and extreme sensitivity to initial conditions are not normal right? What can it mean? Would you suggest another way to estimate the covariance as well?
Thanks a lot again!!
Kyriacos